# Inverse Sine, Cosine, & Tangent - Video Tutorials & Practice Problems

## Inverse Cosine

## Example 1

Evaluate the expression.

$\cos^{-1}\left(-1\right)$

$0$

$\pi$

$\frac{\pi}{2}$

$\frac{3\pi}{2}$

Evaluate the expression.

$\cos^{-1}\left(0\right)$

$0$

$\frac{\pi}{2}$

$\pi$

$\frac{3\pi}{2}$

Evaluate the expression.

$\cos^{-1}\left(-\frac{\sqrt2}{2}\right)$

$\frac{\pi}{4}$

$\frac{3\pi}{4}$

$\frac{5\pi}{4}$

$\frac{7\pi}{4}$

## Inverse Sine

## Example 2

Evaluate the expression.

$\sin^{-1}1$

$0$

$\frac{\pi}{2}$

$\pi$

$-\frac{\pi}{2}$

Evaluate the expression.

$\sin^{-1}\left(\frac{\sqrt3}{2}\right)$

$\frac{\pi}{6}$

$\frac{3\pi}{2}$

$\frac{\pi}{3}$

$-\frac{\pi}{3}$

Evaluate the expression.

$\sin^{-1}\left(\frac{\sqrt2}{2}\right)$

$\frac{\pi}{4}$

$-\frac{\pi}{4}$

$\frac{3\pi}{4}$

$\frac{7\pi}{4}$

## Inverse Tangent

## Example 3

Evaluate the expression.

$\tan^{-1}0$

$0$

$\frac{\pi}{2}$

$\pi$

$-\frac{\pi}{2}$

Evaluate the expression.

$\tan^{-1}1$

$0$

$\frac{\pi}{4}$

$\frac{\pi}{2}$

$-\frac{\pi}{4}$

Evaluate the expression.

$\tan^{-1}\left(-\frac{\sqrt3}{3}\right)$

$\frac{\pi}{6}$

$\frac{\pi}{3}$

$-\frac{\pi}{6}$

$-\frac{\pi}{3}$

Evaluate the expression using a calculator. Express your answer in radians, rounding to two decimal places.

$\tan^{-1}\left(5\right)$

– 5

– 3.38

1.37

78.69

Evaluate the expression using a calculator. Express your answer in radians, rounding to two decimal places.

$\sin^{-1}\left(-\frac13\right)$

0.34

– 0.34

– 0.33

– 19.47

Evaluate the expression using a calculator. Express your answer in radians, rounding to two decimal places.

$\cos^{-1}\left(\frac14\right)$

75.52

1.82

1.32

0.97

## Do you want more practice?

- In Exercises 1–26, find the exact value of each expression. sin⁻¹ 1/2
- In Exercises 1–26, find the exact value of each expression. _ sin⁻¹ √2/2
- Decide whether each statement is true or false. If false, explain why.The tangent and secant functions are und...
- In Exercises 1–26, find the exact value of each expression. sin⁻¹ (- 1/2)
- Which one of the following equations has solution 0?a. arctan 1 = xb. arccos 0 = xc. arcsin 0 = x
- Write each trigonometric expression as an algebraic expression in u, for u > 0.tan (sin⁻¹ u/(√u² + 2)) ...
- Write each trigonometric expression as an algebraic expression in u, for u > 0. ...
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y. Do not use a calculator. y = sec⁻¹ (―2)
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y if it exists. Do not use a calculator.y = sin⁻¹ 0
- Find the exact value of each real number y. Do not use a calculator. y = arccot (―1)
- Find the exact value of each real number y if it exists. Do not use a calculator.y = sin⁻¹ (―1)
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y if it exists. Do not use a calculator.y = cos⁻¹ (―1)
- Give the degree measure of θ. Do not use a calculator.θ = arcsin (―√3/2)
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y if it exists. Do not use a calculator.y = tan⁻¹ 1
- Use a calculator to approximate each value in decimal degrees. θ = arctan 1.7804675
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y if it exists. Do not use a calculator.y = arctan 0
- Use a calculator to approximate each value in decimal degrees.θ = cos⁻¹ 0.80396577
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y if it exists. Do not use a calculator.y = arcsin (―√3/2)
- Use a calculator to approximate each value in decimal degrees.θ = arcsec 3.4723155
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y if it exists. Do not use a calculator.y = arccos (―√3/2)
- Evaluate each expression without using a calculator.cos (arccos (-1))
- Solve each equation for exact solutions. ...
- Find the exact value of each real number y if it exists. Do not use a calculator. y = sin⁻¹ √3
- Evaluate each expression without using a calculator.arccos (cos (3π/4))
- Solve each equation for exact solutions.4/3 cos⁻¹ x/4 = π
- Find the exact value of each real number y if it exists. Do not use a calculator.y = cot⁻¹ (―1)
- Evaluate each expression without using a calculator.tan⁻¹ (tan (π/4))
- Solve each equation for exact solutions.6 sin⁻¹ x = 5π
- Find the exact value of each real number y if it exists. Do not use a calculator.y = csc⁻¹ (―2)
- Evaluate each expression without using a calculator.sin (arccos (3/4))
- Which one of the following equations has solution 3π/4a. arctan 1 = xb. arcsin √2/2 = xc. arccos (―√2 /2) = x
- Solve each equation for exact solutions.cos⁻¹ x = sin⁻¹ 3/5
- Find the exact value of each real number y if it exists. Do not use a calculator.y = arcsec (2√3)/3
- Evaluate each expression without using a calculator.cos (csc⁻¹ (-2))
- Solve each equation for exact solutions.tan⁻¹ x = cot⁻¹ 7/5
- Find the exact value of each real number y if it exists. Do not use a calculator.y = sec⁻¹ 1
- Evaluate each expression without using a calculator.tan (arcsin (3/5) + arccos (5/7))
- Solve each equation for exact solutions.arcsin x = arctan 3/4
- Find the exact value of each real number y if it exists. Do not use a calculator.y = csc⁻¹ √2/2
- Write each trigonometric expression as an algebraic expression in u, for u > 0.tan (arcsec (√1―u²) / u)
- Solve each equation for exact solutions.2 arccos (x/3 - π/3) = 2π
- Find the degree measure of θ if it exists. Do not use a calculator. ...
- Solve each equation for exact solutions.sin⁻¹ x - tan⁻¹ 1 = -π/4
- Find the degree measure of θ if it exists. Do not use a calculator.θ = arcsin (-√3/2)
- The point (π/4, 1) lies on the graph of y = tan x. Therefore, the point _______ lies on the graph of y = tan⁻¹...
- Solve each equation for exact solutions.arccos x + 2 arcsin √3/2 = π
- Find the degree measure of θ if it exists. Do not use a calculator.θ = arccos (-1/2)
- Solve each equation for exact solutions.sin⁻¹ x - 4 tan⁻¹ (-1) = 2π
- Find the degree measure of θ if it exists. Do not use a calculator.θ = cot⁻¹ (-√3/3)
- Solve each equation for exact solutions.arcsin 2x + arccos x = π/6
- Find the degree measure of θ if it exists. Do not use a calculator.θ = csc⁻¹ (-2)
- Solve each equation for exact solutions.cos⁻¹ x + tan⁻¹ x = π/2
- Find the degree measure of θ if it exists. Do not use a calculator.θ = sin⁻¹ 2
- Use a calculator to approximate each value in decimal degrees.θ = sin⁻¹ (-0.13349122)
- Solve each equation for exact solutions. ...
- Which one of the following equations has solution π?a. arccos (―1) = xb. arccos 1 = xc. arcsin (―1) = x
- Find the exact value of each real number y. Do not use a calculator.y = sin⁻¹ √2/2
- The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions...
- Use a calculator to approximate each value in decimal degrees.θ = arccos (-0.39876459)
- Use a calculator to approximate each value in decimal degrees.θ = csc⁻¹ 1.9422833
- Use a calculator to approximate each value in decimal degrees.θ = cot⁻¹ (-0.60724226)
- Solve each equation for x.4/3 arctan x/2 = π
- Solve each equation for x.arccos x + arctan 1 = 11π/12
- Use a calculator to approximate each value in decimal degrees.θ = tan⁻¹ (-7.7828641)
- Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)y = arcsin ...
- Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.) y = cos⁻¹ ...
- Solve each equation for x. y = 1/2 tan (3x + 2), for x in [-2/3 - π/6, -2/3 + π/6]
- Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.) y = arctan...
- Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.) y = cot⁻¹ ...
- Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)y = sec⁻¹ (...
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y. Do not use a calculator.y = tan⁻¹ (―√3)
- Graph each inverse circular function by hand.y = arcsec [(1/2)x]
- Evaluate each expression without using a calculator. ...
- Evaluate each expression without using a calculator. ...
- Evaluate each expression without using a calculator. ...
- Evaluate each expression without using a calculator. ...
- Evaluate each expression without using a calculator.sin (2 cos⁻¹ (1/5))
- Evaluate each expression without using a calculator.sec (sec⁻¹ 2)
- Evaluate each expression without using a calculator.cos (tan⁻¹ (5/12) - tan⁻¹ (3/4))
- Evaluate each expression without using a calculator.sin (sin⁻¹ 1/2 + tan⁻¹ (-3))
- Solve each equation for x, where x is restricted to the given interval. ...
- Find the exact value of each real number y. Do not use a calculator.y = cos⁻¹ (―√2/2)
- Use a calculator to find each value. Give answers as real numbers. ...
- Use a calculator to find each value. Give answers as real numbers. ...
- Write each trigonometric expression as an algebraic expression in u, for u > 0.sin (arccos u)
- Write each trigonometric expression as an algebraic expression in u, for u > 0.cos (arcsin u) ...
- Write each trigonometric expression as an algebraic expression in u, for u > 0.sin (2 sec⁻¹ u/2) ...
- In Exercises 1–26, find the exact value of each expression. _ cos⁻¹ √3/2
- In Exercises 1–26, find the exact value of each expression. _ cos⁻¹ (- √2/2)
- In Exercises 1–26, find the exact value of each expression. _ tan⁻¹ √3/3
- In Exercises 1–26, find the exact value of each expression. _ tan⁻¹ (−√3)
- In Exercises 1–26, find the exact value of each expression. _ cot⁻¹ √3
- In Exercises 1–26, find the exact value of each expression. _ cot⁻¹ (−√3)
- In Exercises 1–26, find the exact value of each expression. _ csc⁻¹ (− 2√3/3)
- In Exercises 1–26, find the exact value of each expression. _ sec⁻¹ (−√2)
- In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻...
- In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻...
- In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. cos⁻...
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ sin⁻¹ (− √3/2)
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. cos⁻¹ (− 1/2)
- In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. _ cos...
- In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. tan⁻...
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sec⁻¹ (−1)
- In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. ___ t...
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ cos(sin⁻¹ √2/2)
- In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ ...
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan[sin⁻¹ (− 1/2)]
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ csc(tan⁻¹ √3/3)
- In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin...
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin(cos⁻¹ 3/5)
- In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan (tan⁻¹...
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan [cos⁻¹ (− 4/5)]
- In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ [tan...
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(sin π/3)
- In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ (tan...
- In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin⁻¹(cos 2π/3)
- In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin...
- In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin(sin⁻¹ ...
- In Exercises 52–53, use a right triangle to write each expression as an algebraic expression. Assume that x is...
- In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and ...
- In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and ...
- In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and ...
- In Exercises 63–82, use a sketch to find the exact value of each expression. cos (sin⁻¹ 4/5)
- In Exercises 63–82, use a sketch to find the exact value of each expression. tan (cos⁻¹ 5/13)
- In Exercises 63–82, use a sketch to find the exact value of each expression. tan [sin⁻¹ (− 3/5)]
- In Exercises 63–82, use a sketch to find the exact value of each expression. _ sin (cos⁻¹ √2/2)
- In Exercises 63–82, use a sketch to find the exact value of each expression. tan [cos⁻¹ (− 1/3)]
- In Exercises 63–82, use a sketch to find the exact value of each expression. cos [tan⁻¹ (− 2/3)]
- In Exercises 63–82, use a sketch to find the exact value of each expression. cot (csc⁻¹ 8)
- In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is...
- In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is...
- In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is...
- The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use trans...
- The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use trans...
- The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use trans...